Archive for September 18th, 2010

Are these the same proof?

September 18, 2010

I have no pressing reason for asking the question I’m about to ask, but it is related to an old post about when two proofs are essentially the same, and it suddenly occurred to me while I was bathing my two-year-old son.

Consider the problem of showing that the product of any $k$ consecutive positive integers (or indeed any integers, not that that is a significant extension) is divisible by $k!.$ I think the proof that most experienced mathematicians would give is the slick one that $n(n+1)\dots(n+k-1)$ divided by $k!$ is $\binom {n+k-1}k,$ and so is the number of ways of choosing $k$ objects from $n+k-1$ objects. Since the latter must be an integer, so must the former.

One might argue that this is not a complete proof because one must show that $\binom{n+k-1}k$ really is the number of ways of choosing $k$ objects from $n+k-1,$ but that is not hard to do.
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